Dissipative systems theory : analysis and synthesis

Abstract

Finite L2 gain and passivity (or positive real) methods have recently played an important role in a large number of robust, high performance engineering designs for both nonlinear and linear systems. This has renewed interest in the classical concept of dissipative systems. In particular, in various finite gain or passivity system synthesis methods in the literature, one studies a relevant dissipation inequality and looks for an appropriate solution to it. When such a solution exists, one then constructs the desired system by using this solution. The main theme of the thesis is the development of a framework for general dissipative systems analysis and synthesis. We firstly present a numerical method for testing dissipativity of a given system. We characterize a dissipative system in terms of a weak (viscosity) solution to a partial differential inequality (PDI) which is the relevant dissipation inequality for the system being considered and develop a finite-difference based discretization method that results in a partial difference inequality approximating the PDI. We then propose two iterative methods to solve the partial difference inequality. We report a number of computational experiment results to demonstrate the utility of the method. Under certain circumstances, strict dissipativity is of the main concern. We provide characterization of a strongly stable, strictly quadratic dissipative nonlinear system in terms of a solution to a PDI or a solution to a partial differential equation (PDE), in the viscosity sense. When the solution to the PDE is smooth, then it also has a stabilizing (in some sense) property. These results generalize the strict bounded real lemma in the linear H control literature. We also provide characterization of a stable, strictly quadratic dissipative linear system in terms of a stabilizing solution to an algebraic Riccati equation (ARE). Connections between quadratic dissipative systems and finite gain related systems are given. In the thesis, we propose a synthesis method for a general dissipative control problem for nonlinear and linear systems with state feedback. We express the solution to the roblem in terms of a solution to a Hamilton-Jacobi-Isaacs (HJI) PDI/PDE in the non­ linear systems case (algebraic Riccati equation/inequality in the linear systems case). In particular, in the case of nonlinear systems with a general quadratic supply rate, we show that whenever there exists a static state feedback control that renders the closed loop system dissipative, then there exists a solution to the Hamilton-Jacobi-Isaacs PDI/PDE in the viscosity solution. This extends and generalizes a number of synthesis results in the nonlinear H control literature. We then consider a general dissipative output feedback control problem and propose a solution by employing the recently developed information state method. We formulate an information state and then convert the original output feedback problem into a new full state one in which the information state provides the appropriate state. The dynamics of the information state takes the form of a controlled PDE. We then solve the new problem by using game theoretic methods leading to a (infinite dimensional) HJI PDI. This is the relevant (ontrolled) dissipation inequality for the output feedback problem at hand. The solution is then specialized to bilinear and linear systems yielding finite dimensional solutions. As a by product, we formulate and solve a general dissipativity filtering problem for nonlinear and linear systems. The problem takes the nonlinear H filtering as a special case. As in the control case, the solution to the filtering problem is expressed in terms of a controlled PDE describing the dynamics of the corresponding information state and a(infinite dimensional) HJI PDI. When specialized to linear systems with a general quadratic supply rate, the solution reduces to new finite dimensional linear filters with the (central) linear H filter appearing as a special one. Finally, we propose application of general dissipativity control methods to two stabilization problems. In the first problem we look for a controller that stabilizes linear systems possesing sector bounded nonlinearities at their inputs and outputs. In the second one, we look for a controller that stabilizes an uncertain nonlinear systenfconsisting of a nonlinear nominal model and an unknown nonlinear model belonging to a class of general dissipative systems described in terms of a specific suppply rate function. In either case, we pose the stabilization problem as a dissipativity control synthesis one for a related system